1918]
RANDOM DISTRIBUTION OF LUMINOUS SOURCES
577
raction of a wave-length. The conclusion that when an exact number of vave-lengths is included the expectation is n remains undisturbed, and this although the effect due to any small part, supposed to act alone, is pro-)ortional to n2. But the influence of any outstanding fraction of a wave-ength is now of increased importance. If we do not look too minutely, the listribution of sources is approximately uniform. If it were" completely so, he whole intensity would be attributable to the fractions at the ends*, and vould be proportional to n2. In general we may expect a part proportional o nz due to the ends and another part proportional to n due to incomplete miformity of distribution over the whole length. When n is small the latter >arb preponderates, but when n is great the situation is reversed, unless the Lumber of wave-lengths included be very nearly integral. And it is apparent hat the n- part has its origin in the discontinuity involved in the sharp imitation of the line, and may be got rid of by a tapering aAvay of the erminal distribution.
Similar ideas are applicable to a random distribution in three dimensions ver a volume, such as a sphere, which may be regarded as composed of chords larallel to the direction in which the effect is to be estimated. The n2 term orresponds to what would be due to a continuous uniform distribution over he volume of the same total source, and it may be regarded as due to the iscontinuity at the surface. In addition there is a term in n, due to the ick of complete uniformity of distribution and issuing from every part of he interior.
Thus far we have been considering the operation of given unit sources, y which in the case of sound is meant centres where a given periodic intro-uction (and abstraction) of fluid is imposed. We now pass to the problem f equal small obstacles distributed at random and under the influence of rimary plane waves. It is easy to recognize that these obstacles act as Bcondary sources, but it is not so obvious that the strength of each source lay be treated as given, without regard to the action of neighbours. apprehend, however, that this assumption is legitimate; in the case of erial waves it may be justified by a calculation upon the -lines of Theory f Sound, § 335. For this purpose we may suppose the density a- of the gas D be unchanged at the obstacles, while the compressibility is altered from m 3 m', so that the secondary disturbance issuing from each obstacle is sym-letrical, of zero order in spherical harmonics. The expressions for the rimary waves and of the disturbance inside the spherical obstacle under onsideration remain as if the obstacle were isolated. But for the secondary isturbance external to the obstacle we must include also that due to neigh-ours. 'On forming the conditions to be satisfied at the surface of the sphere, xpressing the equality on the two sides of pressure (or potential) and of
* It is indifferent how the fraction is divided between the two ends.
R. vi.
37hol on Miss Pockels' plan, behaves quite differently. The first spreading, driving dust to the boundary,